1 Volume visualization I Elvins

surface fitting algorithms • ? marching cubes ? dividing cubes

direct volume rendering algorithms • ? ray casting, integration methods ? voxel projection, projected tetrahedra, splatting

hybrid rendering algorithms • 2 Background & Motivation

Large three-dimensional (3D) data sets arise from measurement • by physical equipment, or from computer simulation.

Scientific areas: computerized tomography (CT), astronomy, • computational physics or chemistry, fluid dynamics, seismology, environmental research, non-destructive testing, etc.

For easy interpretation volume visualization techniques are • useful: ? view data from different viewpoints. ? interactive exploration in Virtual Environments. 3 Requirements

Compression/simplification: visualize reduced version of data • in controllable way.

Progressive refinement: incremental visualization from low to • high resolution.

Progressive transmission: transmit data incrementally from • server to client’s workstation (data transfer is time-limiting factor)

Level-of-detail (LOD): use low resolution for small, distant or • unimportant parts of the data. 4 Volume rendering integral

Transport of light is modelled by equations originating from • physics.

low albedo approximation for the intensity I(x, s) at position • x integrated along the line x + ts, t0 t tl: ≤ ≤

tl t α(x+us)du I(x, s) = Z f(x + ts)e− Rt0 dt. t0

where t0 is the point of entrance, and tl the point of exit. α is the opacity (related to the density of the particles). 5 X-ray rendering

Further simplification: α = 0.

tl I(x, s) = Z f(x + ts)dt. t0 6 Volume visualization II

surface rendering • reduce volume to isosurfaces S(c): f(x, y, z) = c of a density function f(x, y, z) representing the boundary between materials.

direct volume rendering • map volume data directly on screen (no graphical primitives) with semi-transparent effects

forward projection FTB

BTF 7 Direct Volume Rendering

X-ray rendering of human head CT data. 8 X-ray rendering 9 X-ray rendering

integrate the density f along the line of sight • Mathematical concept: X-ray transform: •

Pθf(u, v) = Z f(uu + vv + tθ) dt . R

z v

y θ u

x Pθ (u,v) 10 Voxel vs. Cell model

Represents the 3D division of space by a 3D array of grid points.

gridpoint gridpoint

VOXEL CELL

Voxel: grid point in center, constant value in voxel • Cell: grid points at vertices, value within cell varies • 11 Generalized Voxel model Hohne¨ et al.

3D array with data about intensity values of different materials or information about class membership of certain organs Medical data sources:

MRI soft tissue: fat, muscle

CT hard tissue: bone

PET energy emission, fluid flow, physiology 12 CT vs. MRI

CT image: 512 x 512 pixels, 2 bytes/voxel. 13 CT vs. MRI

MR image: 256 x 256 pixels, 2 bytes/voxel. 14 MRI 15 PET 16 Series of PET scans 17 Orthogonal slices 18 Iso-surface: bone 19 Iso-surface: skin 20 Marching Cubes Lorensen & Cline

1. read 4 successive slices into memory

2. consider cube with 8 data points as vertices : 4 in first slice, 4 in next slice

3. classify vertices 1=inside, 0=outside surface w.r.t. iso-value to determine index of the cube

4. use index to retrieve intersection- and triangulation pattern from look-up table

5. determine exact cube side intersections and surface normals by interpolation of vertex values

6. for each triangle from table, pass the computed 3 vertex values to graphical hardware 21 Marching Cubes: setup

Slice k+1

(i,j+1,k+1) (i+1,j+1,k+1)

(i,j,k+1) (i+1,j,k+1)

(i,j+1,k) (i+1,j+1,k) (i,j,k) (i+1,j,k) Slice k k

j pixel i 22 Marching Cubes

256 possible patterns : by inversion and rotation reduce to 15 basic patterns

0 1 2 3 4

5 6 7 8 9

10 11 12 13 14 23 Marching Cubes: test cube

e v7 v6 6 e v 11 e 3 e 10 v2 2 e e 5 7 e e 3 1 e v4 4 v5 e e 8 9 v0 e v1 0 index v7 v6 v5 v4 v3 v2 v1 v0 24 Marching Cubes: index table

index basic pattern inversion rotation 0 0 0 0, 0, 0 1 1 0 0, 0, 0 2 1 0 1, 0, 0 3 2 0 0, 0, 0 . . . . 255 0 1 0, 0, 0 basic pattern intersection of sides triangulation pattern 0 – – 1 e1e4e9 e1e9e4 2 e2e4e9e10 e2e10e4, e10e9e4 . . . 14 e1e2e6e7e9e11 e1e2e9, e2e7e9, e2e6e7, e9e7e11 25 Cube side intersections

The exact cube side intersections are determined by linear interpolation.

intensity f(1)

c

f(0)

0 x* 1 distance along edge 26 Surface normals

The surface normals are determined by linear interpolation of • vertex normals.

Each vertex normal is computed by central differences: • s(i + 1, j, k) s(i 1, j, k)  − −  Ns = s s(i, j + 1, k) s(i, j 1, k) ∇ ≈ − −  s(i, j, k + 1) s(i, j, k 1)   − −  where s(i, j, k) is the data array. 27 Marching Cubes: (dis)advantages

+ data-volume scanned only once for 1 iso-value

+ determination of geometry independent of viewpoint

– ambiguity in possible triangulation pattern 4 4?

– holes in geometry may arise - 6 3 28 Dividing Cubes

Principle: large numbers of small triangles (projection smaller than pixel) to be mapped as points

look for cells with vertex values not all equal • subdivide cells if projection larger than a pixel • interpolate gradient vector from vertices • map surface point to pixel with computed light intensity • no surface primitives needed; so also no surface-rendering • hardware 29 Surface rendering

Example: Visualization of segmented data (a frog).

Data acquired by physically slicing the frog and photographing the slices.

Data consists of 136 slices of resolution 470 500. × Each pixel labelled with tissue number for a total of 15 tissues. 30 Segmented slice of frog 31 Simplified visualization pipeline

1. read segmentation data

2. remove islands (optional)

3. select tissue by thresholding

4. resample volume (optional)

5. apply Gaussian smoothing filter

6. generate surface using Marching Cubes

7. decimate surface (optional)

8. write surface data 32 Why smooth the volume? 33 Frog 34 Frog Organs 35 3D CT imaging

CT images of bones and tendons of a hand. All tendons and bones are separately segmented. (Courtesy: prof. Frans Zonneveld). 36 Drawbacks of surface rendering

✘ only approximation of surface

✘ only surface means loss of information

✘ amorphous phenomena have no surfaces, e.g. clouds

✘ MR also difficult to visualize: different tissues map to the same scalar value